A filling machine is designed to fill soda bottles with 16 ounces of soda. The distribution for the weight of the bottles is normal. Twenty bottles are selected and weighed. The sample mean is 15.3 ounces and sample standard deviation is 1.5 ounces. Develop a 90% confidence interval for this sample.
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To develop a 90% confidence interval for the sample mean, we can use the formula:
CI = sample mean ± (critical value) * (sample standard deviation / √n)
Where:
- CI is the confidence interval
- sample mean is the mean weight of the sample bottles (15.3 ounces)
- critical value is the value corresponding to the desired confidence level (90%)
- sample standard deviation is the standard deviation of the sample weights (1.5 ounces)
- n is the sample size (20 bottles)
First, we need to find the critical value. For a 90% confidence level, the critical value can be found from the standard normal distribution table or by using a calculator. From the table, the critical value for a 90% confidence level is found to be approximately 1.645.
Now, we can substitute the values into the formula:
CI = 15.3 ± 1.645 * (1.5 / √20)
Calculating the expression within the parentheses:
1.645 * (1.5 / √20) ≈ 0.5457
Substituting the value into the formula:
CI = 15.3 ± 0.5457
Calculating the confidence interval:
Lower limit = 15.3 - 0.5457 ≈ 14.7543
Upper limit = 15.3 + 0.5457 ≈ 15.8543
Therefore, the 90% confidence interval for the sample mean weight of soda bottles is (14.7543, 15.8543) ounces.
To develop a confidence interval for this sample, we need to use the sample mean, sample standard deviation, and the confidence level. In this case, the sample mean is 15.3 ounces and the sample standard deviation is 1.5 ounces. The confidence level is given as 90%.
1. Determine the margin of error:
The margin of error is calculated using the formula:
Margin of Error = Critical Value * (Standard Deviation / √(Sample Size))
2. Find the critical value:
The critical value corresponds to the desired confidence level. Since we want a 90% confidence interval, we need to find the z-value that corresponds to a confidence level of 90%. This can be found using a z-table or a calculator. For a 90% confidence level, the critical value is approximately 1.645.
3. Calculate the margin of error:
Margin of Error = 1.645 * (1.5 / √20) ≈ 0.651
4. Calculate the lower and upper bounds of the confidence interval:
Lower Bound = Sample Mean - Margin of Error
Lower Bound = 15.3 - 0.651 ≈ 14.649
Upper Bound = Sample Mean + Margin of Error
Upper Bound = 15.3 + 0.651 ≈ 15.951
5. Write the confidence interval:
The 90% confidence interval for this sample is (14.649, 15.951) ounces.
Please note that this confidence interval means we are 90% confident that the true population mean falls within this range based on our sample data.